A206435 Total sum of odd parts in the last section of the set of partitions of n.

1, 1, 5, 3, 13, 13, 29, 29, 66, 70, 126, 146, 241, 287, 450, 526, 791, 963, 1360, 1660, 2312, 2810, 3799, 4649, 6158, 7528, 9824, 11962, 15393, 18773, 23804, 28932, 36413, 44093, 54953, 66419, 82085, 98929, 121469, 145865, 177983, 213241, 258585, 308861

Offset: 1

Author

Omar E. Pol, Feb 12 2012

Keywords

• nonn

Comments

From Omar E. Pol, Apr 09 2023: (Start)

Convolution of A002865 and A000593.

a(n) is also the total sum of odd divisors of the terms in the n-th row of the triangle A336811.

a(n) is also the sum of odd terms in the n-th row of the triangle A207378.

a(n) is also the sum of odd terms in the n-th row of the triangle A336812. (End)

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)

Crossrefs

Partial sums give A066967.

Cf. A000593, A002865, A135010, A138121, A138879, A206433, A206434, A206436, A207378, A336811, A336812.

Programs

Maple

b:= proc(n, i) option remember; local g, h;
      if n=0 then [1, 0]
    elif i<1 then [0, 0]
    else g:= b(n, i-1); h:= `if`(i>n, [0, 0], b(n-i, i));
         [g[1]+h[1], g[2]+h[2] +(i mod 2)*h[1]*i]
      fi
    end:
a:= n-> b(n, n)[2] -`if`(n=1, 0, b(n-1, n-1)[2]):
seq(a(n), n=1..60);  # Alois P. Heinz, Mar 16 2012

Mathematica

b[n_, i_] := b[n, i] = Module[{g, h}, Which[n == 0, {1, 0}, i < 1, {0, 0}, True, g = b[n, i-1]; h = If[i > n, {0, 0}, b[n-i, i]]; {g[[1]] + h[[1]], g[[2]] + h[[2]] + Mod[i, 2]*h[[1]]*i}]]; a[n_] := b[n, n][[2]] - If[n == 1, 0, b[n-1, n-1][[2]]]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Feb 16 2017, after Alois P. Heinz *)

Formula

G.f.: (Sum_{i>=0} (2*i+1)*x^(2*i)*(1-x)/(1-x^(2*i+1))) / Product_{j>0} (1-x^j). - Alois P. Heinz, Mar 16 2012

a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (24*sqrt(2*n)). - Vaclav Kotesovec, May 29 2018

Extensions

More terms from Alois P. Heinz, Mar 16 2012